4 3 problem solving angle relationships in triangles answers

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Angle Relationships Simply Explained w/ 11+ Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s lesson, you’re going to learn all about angle relationships and their measures.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’ll walk through 11 step-by-step examples to ensure mastery.

Let’s dive in!

Angle Pair Relationship Names

In Geometry , there are five fundamental angle pair relationships:

• Complementary Angles
• Supplementary Angles
• Adjacent Angles
• Linear Pair
• Vertical Angles

1. Complementary Angles

Complementary angles are two positive angles whose sum is 90 degrees.

For example, complementary angles can be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two acute angles, like ∠MNP and ∠EFG, whose sum is equal to 90 degrees. Both of these graphics represent pairs of complementary angles.

Complementary Angles Example

2. Supplementary Angles

Supplementary angles are two positive angles whose sum is 180 degrees.

For example, supplementary angles may be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two angles, like ∠MNP and ∠KLR, whose sum is equal to 180 degrees. Both of these graphics represent pairs of supplementary angles.

Supplementary Angles Example

What is important to note is that both complementary and supplementary angles don’t always have to be adjacent angles.

3. Adjacent Angles

Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.

Angles 1 and 2 are adjacent angles because they share a common side.

Adjacent Angles Examples

And as Math is Fun so nicely points out, a straightforward way to remember Complementary and Supplementary measures is to think:

C is for Corner of a Right Angle (90 degrees) S is for Straight Angle (180 degrees)

Now it’s time to talk about my two favorite angle-pair relationships: Linear Pair and Vertical Angles.

4. Linear Pair

A linear pair is precisely what its name indicates. It is a pair of angles sitting on a line! In fact, a linear pair forms supplementary angles.

Because, we know that the measure of a straight angle is 180 degrees, so a linear pair of angles must also add up to 180 degrees.

∠ABD and ∠CBD form a linear pair and are also supplementary angles, where ∠1 + ∠2 = 180 degrees.

Linear Pair Example

5. Vertical Angles

Vertical angles are two nonadjacent angles formed by two intersecting lines or opposite rays.

Think of the letter X. These two intersecting lines form two sets of vertical angles (opposite angles). And more importantly, these vertical angles are congruent.

In the accompanying graphic, we see two intersecting lines, where ∠1 and ∠3 are vertical angles and are congruent. And ∠2 and ∠4 are vertical angles and are also congruent.

Vertical Angles Examples

Together we are going to use our knowledge of Angle Addition, Adjacent Angles, Complementary and Supplementary Angles, as well as Linear Pair and Vertical Angles to find the values of unknown measures.

Angle Relationships – Lesson & Examples (Video)

• Introduction to Angle Pair Relationships
• 00:00:15 – Overview of Complementary, Supplementary, Adjacent, and Vertical Angles and Linear Pair
• Exclusive Content for Member’s Only
• 00:06:29 – Use the diagram to solve for the unknown angle measures (Examples #1-8)
• 00:19:05 – Find the measure of each variable involving Linear Pair and Vertical Angles (Examples #9-12)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Triangle Angle Calculator

How to find the angle of a triangle, sum of angles in a triangle - triangle angle sum theorem, exterior angles of a triangle - triangle exterior angle theorem, angle bisector of a triangle - angle bisector theorem, finding missing angles in triangles - example.

Triangle angle calculator is a safe bet if you want to know how to find the angle of a triangle. Whether you have three sides of a triangle given, two sides and an angle or just two angles, this tool is a solution to your geometry problems. Below you'll also find the explanation of fundamental laws concerning triangle angles: triangle angle sum theorem, triangle exterior angle theorem, and angle bisector theorem. Read on to understand how the calculator works, and give it a go - finding missing angles in triangles has never been easier!

There are several ways to find the angles in a triangle, depending on what is given:

• Given three triangle sides

Use the formulas transformed from the law of cosines:

For the second angle we have:

And eventually, for the third angle:

• Given two triangle sides and one angle

If the angle is between the given sides, you can directly use the law of cosines to find the unknown third side, and then use the formulas above to find the missing angles, e.g. given a,b,γ:

• calculate c = a 2 + b 2 − 2 a b × cos ⁡ ( γ ) c = \sqrt{a^2 + b^2 - 2ab \times \cos(\gamma)} c = a 2 + b 2 − 2 ab × cos ( γ ) ​ ;
• substitute c c c in α = a r c c o s ( ( b 2 + c 2 − a 2 ) / ( 2 b c ) ) \alpha = \mathrm{arccos}\left((b^2 + c^2- a^2)/(2bc)\right) α = arccos ( ( b 2 + c 2 − a 2 ) / ( 2 b c ) ) ;
• then find β \beta β from triangle angle sum theorem: β = 180 ° − α − γ \beta = 180\degree- \alpha - \gamma β = 180° − α − γ

If the angle isn't between the given sides, you can use the law of sines. For example, assume that we know a a a , b b b , and α \alpha α :

• As you know, the sum of angles in a triangle is equal to 180 ° 180\degree 180° . From this theorem we can find the missing angle: γ = 180 ° − α − β \gamma = 180\degree- \alpha - \beta γ = 180° − α − β .
• Given two angles

That's the easiest option. Simply use the triangle angle sum theorem to find the missing angle:

• α = 180 ° − β − γ \alpha = 180\degree- \beta - \gamma α = 180° − β − γ ;
• β = 180 ° − α − γ \beta= 180\degree- \alpha - \gamma β = 180° − α − γ ; and
• γ = 180 ° − α − β \gamma = 180\degree- \alpha- \beta γ = 180° − α − β

In all three cases, you can use our triangle angle calculator - you won't be disappointed.

🙋 Meet the law of sines and cosines at our law of cosines calculator and law of sines calculator ! Everything will be clear afterward. 😉

The theorem states that interior angles of a triangle add to 180 ° 180\degree 180° :

How do we know that? Look at the picture: the angles denoted with the same Greek letters are congruent because they are alternate interior angles. Sum of three angles α \alpha α β \beta β , γ \gamma γ is equal to 180 ° 180\degree 180° , as they form a straight line. But hey, these are three interior angles in a triangle! That's why α + β + γ = 180 ° \alpha + \beta+ \gamma = 180\degree α + β + γ = 180° .

An exterior angle of a triangle is equal to the sum of the opposite interior angles .

• Every triangle has six exterior angles (two at each vertex are equal in measure).
• The exterior angles, taken one at each vertex, always sum up to 360 ° 360\degree 360° .
• An exterior angle is supplementary to its adjacent triangle interior angle.

Angle bisector theorem states that:

An angle bisector of a triangle angle divides the opposite side into two segments that are proportional to the other two triangle sides.

Or, in other words:

The ratio of the B D ‾ \overline{BD} B D length to the D C ‾ \overline{DC} D C length is equal to the ratio of the length of side A B ‾ \overline{AB} A B to the length of side A C ‾ \overline{AC} A C :

OK, so let's practice what we just read. Assume we want to find the missing angles in our triangle. How to do that?

• Find out which formulas you need to use . In our example, we have two sides and one angle given. Choose angle and 2 sides option.
• Type in the given values . For example, we know that a = 9   i n a = 9\ \mathrm{in} a = 9   in , b = 14   i n b = 14\ \mathrm{in} b = 14   in , and α = 30 ° \alpha = 30\degree α = 30° . If you want to calculate it manually, use law of sines:
• From the theorem about sum of angles in a triangle, we calculate that γ = 180 ° − α − β = 180 ° − 30 ° − 51.06 ° = 98.94 ° \gamma = 180\degree- \alpha - \beta = 180\degree- 30\degree - 51.06\degree= 98.94\degree γ = 180° − α − β = 180° − 30° − 51.06° = 98.94° .
• The triangle angle calculator finds the missing angles in triangle . They are equal to the ones we calculated manually: β = 51.06 ° \beta = 51.06\degree β = 51.06° , γ = 98.94 ° \gamma = 98.94\degree γ = 98.94° ; additionally, the tool determined the last side length: c = 17.78   i n c = 17.78\ \mathrm{in} c = 17.78   in .

Reasoning similar to the one we applied in this calculator appears in other triangle calculations, for example the ones we use in the ASA triangle calculator and the SSA triangle calculator !

How do I find angles in a triangle?

To determine the missing angle(s) in a triangle, you can call upon the following math theorems:

• The fact that the sum of angles is a triangle is always 180° ;
• The law of cosines ; and
• The law of sines .

Which set of angles can form a triangle?

Every set of three angles that add up to 180° can form a triangle. This is the only restriction when it comes to building a triangle from a given set of angles.

Why can't a triangle have more than one obtuse angle?

This is because the sum of angles in a triangle is always equal to 180° , while an obtuse angle has more than 90° degrees. If you had two or more obtuse angles, their sum would exceed 180° and so they couldn't form a triangle. For the same reason, a triangle can't have more than one right angle!

How do I find angles of the 3 4 5 triangle?

Let's denote a = 5 , b = 4 , c = 3 .

• Write down the law of cosines 5² = 3² + 4² - 2×3×4×cos(α) . Rearrange it to find α , which is α = arccos(0) = 90° .
• You can repeat the above calculation to get the other two angles.
• Alternatively, as we know we have a right triangle, we have b/a = sin β and c/a = sin γ .
• Either way, we obtain β ≈ 53.13° and γ ≈ 36.87 .
• We quickly verify that the sum of angles we got equals 180° , as expected.

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Solving Triangles

"Solving" means finding missing sides and angles.

Six Different Types

If you need to solve a triangle right now choose one of the six options below:

Which Sides or Angles do you know already? (Click on the image or link)

... or read on to find out how you can become an expert triangle solver :

Your Solving Toolbox

Want to learn to solve triangles?

Imagine you are " The Solver " ... ... the one they ask for when a triangle needs solving!

In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:

1. Angles Add to 180° :

A + B + C = 180°

When you know two angles you can find the third.

2. Law of Sines (the Sine Rule):

When there is an angle opposite a side, this equation comes to the rescue.

Note: angle A is opposite side a, B is opposite b, and C is opposite c.

3. Law of Cosines (the Cosine Rule):

This is the hardest to use (and remember) but it is sometimes needed to get you out of difficult situations.

It is an enhanced version of the Pythagoras Theorem that works on any triangle.

Six Different Types (More Detail)

There are SIX different types of puzzles you may need to solve. Get familiar with them:

This means we are given all three angles of a triangle, but no sides.

AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.

We need to know at least one side to go further. See Solving "AAA" Triangles .

This mean we are given two angles of a triangle and one side, which is not the side adjacent to the two given angles.

Such a triangle can be solved by using Angles of a Triangle to find the other angle, and The Law of Sines to find each of the other two sides. See Solving "AAS" Triangles .

This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles.

In this case we find the third angle by using Angles of a Triangle , then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles .

This means we are given two sides and the included angle.

For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle. See Solving "SAS" Triangles .

This means we are given two sides and one angle that is not the included angle.

In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side. See Solving "SSA" Triangles .

This means we are given all three sides of a triangle, but no angles.

In this case, we have no choice. We must use The Law of Cosines first to find any one of the three angles, then we can use The Law of Sines (or use The Law of Cosines again) to find a second angle, and finally Angles of a Triangle to find the third angle. See Solving "SSS" Triangles .

Tips to Solving

Here is some simple advice:

When the triangle has a right angle, then use it, that is usually much simpler.

When two angles are known, work out the third using Angles of a Triangle Add to 180° .

Try The Law of Sines before the The Law of Cosines as it is easier to use.

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Course: 7th grade   >   Unit 9

• Find measure of vertical angles
• Finding missing angles
• Find measure of angles word problem
• Equation practice with complementary angles
• Equation practice with supplementary angles
• Equation practice with vertical angles
• Create equations to solve for missing angles

Unknown angle problems (with algebra)

• Your answer should be
• an integer, like 6 ‍  
• an exact decimal, like 0.75 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  

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Angle Relationships In Triangles

In this video, we are going to look at the angle relationships in a triangle.

Video-Lesson Transcript

In this lesson, we’ll cover angle relationships in a triangle.

A triangle has three angles.

If we add this all up:

Let’s give values to this:

This is also because:

This is true for any of these.

And the exterior angles is equal to the sum of the other two interior angles.

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Triangle Calculator

Enter the values of any two angles and any one side of a triangle below which you want to solve for remaining angle and sides.

Triangle calculator finds the values of remaining sides and angles by using Sine Law.

Sine law states that

a sin A = b sin B = c sin C

Cosine law states that-

a 2 = b 2 + c 2 - 2 b c . cos ( A ) b 2 = a 2 + c 2 - 2 a c . cos ( B ) c 2 = a 2 + b 2 - 2 a b . cos ( C )

Click the blue arrow to submit. Choose "Solve the Triangle" from the topic selector and click to see the result in our Trigonometry Calculator!

Solve the Triangle

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